Need to sort by matching columns

I have a two spreadsheets with a lot of data on them that I am trying to make sure have been transcribed correctly from one to the other through manual entry. I suspect this is not the case and I want to identify the errors and fix them. I took the relevant columns from each sheet and put them next to each other on a new one. Column A shows part number and Column B the corresponding quantity. Column C shows part number from the second sheet, and again Column D has the corresponding quantity. Unfortunately, there is a vast discrepancy in number of lines, as Column A has many more part numbers in it than Column C does.

What I am attempting to accomplish is to have Excel find any exact matches between Columns A and C, and sort by that, while keeping the correct quantity data from Column B on the same row with its corresponding part number in Column A, and likewise for Columns D & C. Ideally, any rows that did not have a match, or where Column C was blank would be sorted to the bottom of the sheet.

Attached is a small piece of the sheet I am dealing with for simplicity's sake. The full file has more than 10,000 rows. Any help would be greatly appreciated. Thank you.